Problem: What is the average rate of change of the function $f(x)=\log(x)$ over the interval $[5,5+h]$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{\log(5+h)-\log(5)}{h}$ (Choice B) B $\dfrac{\log(h)-\log(5)}{h}$ (Choice C) C $\dfrac{\log(5+h)-\log(5)}{5+h}$ (Choice D) D $\dfrac{\log(5+h)-\log(5)}{5}$
Answer: This is the formula for the average rate of change of a function $f$ over the interval $[a,b]$ : $\dfrac{f(b)-f(a)}{b-a}$ We are interested in the average rate of change of $f(x)=\log(x)$ over the interval $[5,5+h]$ : $\begin{aligned} &\phantom{=}\dfrac{f(5+h)-f(5)}{(5+h)-(5)} \\\\ &=\dfrac{\log(5+h)-\log(5)}{5+h-5} \\\\ &=\dfrac{\log(5+h)-\log(5)}{h} \end{aligned}$ The average rate of change of the function is $\dfrac{\log(5+h)-\log(5)}{h}$. Notice that the average rate of change is calculated just like the slope of the secant line that intersects the graph of the function at the interval's endpoints.